Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,16)(18,20)(21,22,24,26)(23,27,28,25);(1,2,4,3)(5,7,10,9)(6,11,8,12)(15,19)(18,20)(21,22,24,26)(23,25,28,27);(2,3)(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,19,20)(21,23)(22,25)(24,28)(26,27);(5,8,10,6)(7,11,9,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(1,3,4,2)(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,20,19,18)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=ghi,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300890, #72300908, #72300910, … |
| maximal normal subgroup of | #72300890, #72300908, #72300910, … |
| maximal quotient of | #72330089, #72331424 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2475236);